Kinematics and Dynamics Kinematics and Dynamics Physics Help Forum 
Apr 23rd 2014, 09:41 AM

#1  Junior Member
Join Date: Dec 2012
Posts: 24
 Ranking the Velocity and Acceleration of an Object
Greetings,
I have a unique problem regarding the acceleration and velocity of an object. Previously, I submitted some questions to this forum in regards to sprinters and the time v. distance derivatives thereof. This time, I have a new angle regarding a similar question.
SITUATION:
A fastest runner runs 40 yards in a straight line. From the starting line (0 yds) to the 10 yard line (10 yards) segement, he runs it in 1.40 seconds. From the 10 yard line (10 yards) to the finish line (40 yards), he runs this segment in 2.69 seconds. His total time to run 40 yards is thus 1.40 + 2.69 = 4.09 seconds.
A talent scout determines that his acceleration phase ended at the 10 yard marker and his velocity stage (where acceleration = 0) was from the 10 yard marker to the 40 yard marker. In the acceleration phase, his acceleration was constant. In the velocity phase, his velocity was constant.
The talent scout determines that this fastest runner gets a score for each of the two phases. The first segment from 0 to 10 yards was ran in 1.40 seconds. The talent scout gives the runner a grade of 99/99 for this phase. The segment from 10 to 40 yards was ran in 2.69 seconds. The talent scout gives the runner a grade of 99/99 for this phase as well. His total score for both phases is 198/198. This is the highest score possible because the runner achieved the highest score possible in each segment.
Now, the talent scout is going to evaluate other SLOWER runners who have various times for each of their two segments. Assume that in each phase the velocity or acceleration is constant like in the example of the fastest runner. After recording his data, the talent scout determines that the SLOWEST runner had a time of 2.19 seconds from 010 yards and a time of 4.24 seconds from 1040 yards. This gave the slowest runner a total time from 040 yards of 6.43 seconds. The talent scout assigns a grade of 1/99 for the first segment traversed in 2.19 seconds and a score of 1/99 for the second segment ran in 4.24 seconds. The slowest runner's total score is 2/198, which is the lowest score possible.
The talent scout, wanting to evaluate other runners wants to create a grading scale to determine each runner's grade for their acceleration and velocity phases useing their split times from each of their two segments (010 and 1040 yds). He sets the fastest acceleration phase time of 1.40 equal to 99 (1.40 = 99) and the slowest acceleration phase time of 2.19 equal to 1 (2.19 = 1). He determines that the linear equation between these two points, solving for y (grade) dependent on x (segment time) is y = 124.05x + 272.67. Therefore, a runner with an acceleration split time of 1.50 seconds will get a grade of 87/99 for their acceleration phase. The talent scout also sets the fastest velocity phase (040 yds) of 2.69 seconds equal to 99 (2.69 = 99) and the slowest velocity phase of 4.24 equal to 1 (4.24 = 1). The linear equation he derives between these two points, is y = 63.226x + 269.08. Therefore, a runner with a velocity phase segment time of 3.00 seconds receives a grade of 79/99 for the velocity phase.
This process seems to work great for the talent scout who wants to see what the total scores for two runners with the same 040 yard time will be. Runner A has a first phase time of 1.50 seconds. This gives him an acceleration phase score of 87. Runner A also has a velocity phase time of 2.93 seconds, which gives Runner A a second phase grade of 84. His overall scores is 171 (87 + 84) and his overall time is 4.43 seconds (1.50 + 2.93).
Runner B has an acceleration phase time of 1.58 seconds. This gives him an acceleration phase score of 77. Runner B has a velocity phase time of 2.85 seconds giving him a velocity phase score of 89. Runner B's total time for the 40 yard run was also 4.43 seconds (1.58 + 2.85), the same time as Runner A. Naturally, the talent scout estimates that the total scores must also be the same considering that both runners ran the same time over the course of 40 yards, despite the fact that Runner A had a better acceleration phase and Runner B had a better velocity phase. However, when the talent scout added up both scores, Runner A had a total score of 171 (87 + 84) but Runner B had a total score of 166 (77 + 89).
QUESTION:
How can the talent scout adjust his system so when runners achieve the same time over the course of the entire 40 yards, they will have the same overall grade out of 198? In other words, is it possible for Runner A and Runner B to have different segement times and still have the same score? If a runner has grades of 90 + 99 = 189 and another has grades of 99 + 90 = 189 and they ran the same time, this makes sense. But in the example of the two runners above, how can the talent scout's system be adjusted to reflect that both players have the same overall score (out of 198) and same overall time despite different segment scores? We know that both runners ran the same overall time of 4.43 seconds, but yet their scores are different (171 to 166). Is there a way to get the scores to match up or is this a biproduct of using a linear ranking system?
Any adivce on how I can solve this problem would be GREATLY appreciated. Thank you all again for your help!
DB

 
Apr 23rd 2014, 11:46 AM

#2  Physics Team
Join Date: Jun 2010 Location: Morristown, NJ USA
Posts: 2,347

First  something is wrong with your model, or at least the hypothetical results that you gave are inconsistent with the model. You have said to assume constant acceleration in the first 10 yards and constant velocity for the next 30 yards. If this is the case then the runner with the highest score in the acceleration phase must also achieve the highest score in the velocity phase. This is because the person with the fastest time in the accaleration phase must have the highest acceleration and so also achieves the fastest velocity at the 10 yard mark, and if velocity is constant in phase 2 then he will maintain the fastest veocity for the entire second phase. In fact we can predict the time for the velocity phase purely from the time for the acceleration phase. The math is like this:
For phase 1: d_1 = (1/2)at_1^2, so a= 2d_1/t_1^2. His velocity at the end of the acceleration phase is v=at_1 = 2d_1/t_1. His time to run the next 30 yards is then t_2 = d_2/v = d_2 t_1/(2 d_1). Since d_2/d_1 = 3, this becomes t_2 = 1.5t_1.
Since your data doesn't fit this, the model is incorrect. But all may not be lost. Your approach is to weigh both phase's equally, but there is much more variation in times for the 2nd phase than the first, because it takes longer to complete on average, and so your method overweights small differences in phase 1 results compared to phase 2. That's why two runner's with same total times get different scores in your system.
If your goal is to have equal total scores for equal total times let's start with that and work backwards. First calculate the mean of times for each phase and for the total times  call them u_1, u_2 and u_t. For each runner then calculate for each phase the difference from the runner's time for that phase and the mean time, and divide by the mean of the total time. Thus if t_1 and t_2 are a runner's times for the two phases, his score for phase 1 is (t_1u_1)/u_t and the score for phase 2 is (t_2u_2)/u_t. The total score is the sum, and it works out to (t_tu_t)/u_t. Note that the total score depends only on total time; it's not weighted by phase. These scores will be on the order of .3 to +.3. If you want to assign a 99point scale to each phase and the total that's fine, but these 99point scales aren't additive.
Using your data I get the following:
Runner A: Phase 1 = 79.2, Phase 2 = 83.7, total score = 82.6
Runner B: Phase 1 = 63.4, Phase 2 = 88.8, total score = 82.6
Note that Runner B scored 16 points less than Runner A on phase 1, and only 5 points better on phase 2, but because phase 2 is more important in the overall results that allowed him to tie runner A for his total score.
One more thought  if you want the 99percent scores to be able to add, you need to weight phase 1 to be worth less than phase 2. Since the median for phase 1 dicvide by the median for phase 2 is 52%, you could use a 52 point scale for phase 1 and a 99 point scale for phase 2. Then the total scores will be from 2 to 151, and both runners A and B get a total score of 128.
Last edited by ChipB; Apr 23rd 2014 at 11:48 AM.

 
Apr 23rd 2014, 12:39 PM

#3  Junior Member
Join Date: Dec 2012
Posts: 24

Originally Posted by ChipB First  something is wrong with your model, or at least the hypothetical results that you gave are inconsistent with the model. You have said to assume constant acceleration in the first 10 yards and constant velocity for the next 30 yards. If this is the case then the runner with the highest score in the acceleration phase must also achieve the highest score in the velocity phase. This is because the person with the fastest time in the accaleration phase must have the highest acceleration and so also achieves the fastest velocity at the 10 yard mark, and if velocity is constant in phase 2 then he will maintain the fastest veocity for the entire second phase. In fact we can predict the time for the velocity phase purely from the time for the acceleration phase. The math is like this:
For phase 1: d_1 = (1/2)at_1^2, so a= 2d_1/t_1^2. His velocity at the end of the acceleration phase is v=at_1 = 2d_1/t_1. His time to run the next 30 yards is then t_2 = d_2/v = d_2 t_1/(2 d_1). Since d_2/d_1 = 3, this becomes t_2 = 1.5t_1.
Since your data doesn't fit this, the model is incorrect. But all may not be lost. Your approach is to weigh both phase's equally, but there is much more variation in times for the 2nd phase than the first, because it takes longer to complete on average, and so your method overweights small differences in phase 1 results compared to phase 2. That's why two runner's with same total times get different scores in your system.
If your goal is to have equal total scores for equal total times let's start with that and work backwards. First calculate the mean of times for each phase and for the total times  call them u_1, u_2 and u_t. For each runner then calculate for each phase the difference from the runner's time for that phase and the mean time, and divide by the mean of the total time. Thus if t_1 and t_2 are a runner's times for the two phases, his score for phase 1 is (t_1u_1)/u_t and the score for phase 2 is (t_2u_2)/u_t. The total score is the sum, and it works out to (t_tu_t)/u_t. Note that the total score depends only on total time; it's not weighted by phase. These scores will be on the order of .3 to +.3. If you want to assign a 99point scale to each phase and the total that's fine, but these 99point scales aren't additive.
Using your data I get the following:
Runner A: Phase 1 = 79.2, Phase 2 = 83.7, total score = 82.6
Runner B: Phase 1 = 63.4, Phase 2 = 88.8, total score = 82.6
Note that Runner B scored 16 points less than Runner A on phase 1, and only 5 points better on phase 2, but because phase 2 is more important in the overall results that allowed him to tie runner A for his total score.
One more thought  if you want the 99percent scores to be able to add, you need to weight phase 1 to be worth less than phase 2. Since the median for phase 1 dicvide by the median for phase 2 is 52%, you could use a 52 point scale for phase 1 and a 99 point scale for phase 2. Then the total scores will be from 2 to 151, and both runners A and B get a total score of 128. 
Thanks Chip! Perhaps I did not explain the model correctly, then. I guess the velocity is not constant, then. I think I was overthinking. Basically, the runs are in two segments. Segment 1 from 0 to 10 yards is used to calculate the acceleration rating of a runner while the segment from 1040 yards is used to calculate his velocity or 'speed' rating.
The way the 'actual' model works is that a runner with a 90 acceleration and 99 speed rating will reach the 40 yard mark at the same time as a runner with an acceleration rating of 99 and speed rating of 90. To me, that means that if you add them up, they will get you the same result. So, if Runner A runs segment 1 in 1.5 seconds and Runner B runs segment 1 in 1.58 seconds, the model states that Runner A is the better 'accelerator' of the two. However, since Runner A runs the final 30 yards in 2.93 seconds compared to Runner B who ran that second segment in 2.85 seconds, we can say that Runner B has a better 'Speed' rating.
The model also dictates that each rating is on a 199 scale. So what I am trying to do is determine what each runner's rating for 'Acceleration' and 'Speed' should be so that if you have runners with the same t_t but different t_1 and t_2, they will still reach the 40 yard mark at the same time.
The one thing I neglected was that the first phase is only 25% of the current run, however. So a runner with slow acceleration but faster speed will catch the runner who accelerated quicker but isn't as fast at top speed over the course of the final 30 yards.
Here are the actual u values of the entire population:
u_1 = 1.66
u_2 = 3.15
u_t = 4.81
I tried to use the example I provided with two different runners using your method but I got different results for the segment scores. Can you please walk me through the math to make sure I am doing it correctly to get the scores for Runner A and Runner B? Here are their segment results again.
Runner A:
t_1 = 1.50
t_2 = 2.93
t_t = 4.43
Runner B:
t_1 = 1.58
t_2 = 2.85
t_t = 4.43
I want to state that it DOES make sense that you simply cannnot add them up considering that phase 1 is only 25% of the run. What I really want is a way to accurately predict the ACC and SPD ratings for any runner given their phase 1 and phase 2 segment times so that if they run 40 yards, despite different phase times, they could hypothetically cross the finish line at the 40 yard mark at the same time. Like I said earlier, in the model, a runner with an ACC of 99 and SPD of 90 will cross the 40 yard mark at the same time as a runner with an ACC of 90 and a SPD of 99....in theory.

 
Apr 23rd 2014, 02:25 PM

#4  Physics Team
Join Date: Jun 2010 Location: Morristown, NJ USA
Posts: 2,347

As I said earlier  it's impossible using a linear scale for two runners that have different phase 1 times and phase 2 times but equal total times to also have scores on your 99 point scale add to the same number. Just can't happen, sorry. You must weight the phases according to their value in contributing to the runner's overall result. I suggested that the first phase be rated on a scale of 152 and the 2nd phase on a scale of 199  this would make the math work, but means the max possible score is 99+52=151.
Here's how the math works using your data. I found an error in my spreadsheet which I have fixed, so the numbers below are a bit different than in my previous post. There were four runners  the fast runner, the slow runner, and runners A and B. The data is:
Phase 1:
Fast: 1.4s
A: 1.5s
B:1.68s
Slow:2.19s
u_1 = 1.667s
Phase 2:
Fast: 2.69s
A: 2.93s
B: 2.85s
Slow: 4.24s
u_2=3.178s
Total:
Fast: 4.09s
A: 4.43
B: 4.43s
Slow:6.43s
u_2=4.845s
Phase 1 Raw Scores = (u_1t_1)/u_t
Fast: 0.055
A: 0.035
B: 0.018
Slow: 0.108
(Note that the raw scores sum to zero, as expected)
Phase 1 Scaled scores on 199 scale:
Fast: 99
A: 86.5
B: 76.4
Slow: 1
Phase 2 Raw scores = (u_2t_2)/u_t:
Fast: 0.101
A: 0.051
B: 0.068
Slow: 0.219
Phase 2 Scaled scores on 199 scale:
Fast: 99
A: 83.7
B: 88.8
Slow: 1
Total Raw Scores (u_tt_t)/u_t:
Fast: 0.156
A: 0.086
B: 0.086
Slow: 0.327
Total Scaled score on 199 scale:
Fast: 99
A: 84.62
B: 84.62
Slow: 1
This gives same total scores to A and B, but note that they are not the simple addition of scores for phases 1 and 2. An alternative: add scaled phase 1 and phase 2 scores, with phase 1 weighted at 52.5%:
Fast: 151
A: 129
B: 129
Slow: 1.52
This gives a total score range of 1151, and perhaps better illustrates that phase 2 results are more important than phase 1. It points out to the runner that improving by 0.1 second in phase 2 is better than improving by 0.1 seconds in phase 1. Hope this helps!

 
Apr 23rd 2014, 03:57 PM

#5  Junior Member
Join Date: Dec 2012
Posts: 24

Thank you so much again Chip! This definitely helps me identify the flaws I had in my method!
I am also trying to see what method may be best for my analysis. Here is what I am trying to do, so if you have any ideas on the best way to do this, you can get an idea of where I am coming from.
Basically, I have data points for thousands of football players who run 40 yard dashes. This allows scouts to help determine how well a football player accelerates and how fast they move at top speed, given perfect, straightline, conditions.
The data is measured in seconds and yards with data points at the 10yd, 20yd, and 40yd marks. What I am trying to do is determine how well a player accelerates and how fast they move using this data. It works because the conditions are identical (timing basis and distance) for all players who participate. This allows you to compare one player to another. What I then do is take this data and rate/grade/scale it on a scale of 1 to 99 with 99 being the best and 1 being the worst.
Check out the following video. In this video, both players in the Madden video game run on the field. They both start at the same time and reach the 40 yard mark at the same time. https://www.facebook.com/video/video.php?v=738382887866
The bottom player has an ACC rating of 99, evident by his great initial burst where he accelerates away from the top player. The top player has an ACC of 90. However, the top player being to catch and then pull away from the bottom player because his SPD rating is 99 while the bottom player's speed rating is 90. It breaks down like this:
Bottom Player: ACC = 99, SPD = 90.
Top Player: ACC = 90, SPD = 99.
As you can see in the video they both reach the 40 yard mark (at the 50 yard line because they start at the 10 yard line) at the same moment. From this point forward, the acceleration for both players is equal to 0 and the velocity is constant, evident by the Top Player's ability to continue to pull away.
I want to see how I can rate other players with other split times based on this model or something like it. In a previous forum topic, you told me that I have too few data points to create anything of real substance. You recommended that I use the 10yd split time to measure the ACC rating and the 4010 time to measure the velocity rating.
One thing I considered was finding the average acceleration and velocity for each segment that I have split times for (010yds, 1020yds, and 2040yds). Then, I can determine the sum of the velocities and accelerations and find the percentage of the total used in each phase. Using that I can try to interpolate an overall ACC and SPD rating.
However, I do not know if this is sound logic or possible. I really need to get down to two numbers that are whole number integers from 1 to 99 for both ACC and SPD.
Would you mind looking at the video and seeing if there is a way that I can attempt to at least mimic how Madden rates the SPD and ACC ratings? Also, can you possibly suggest a way for me to convert reallife 40 yard dash splits into those ratings?
Here is a helpful breakdown of the frames and times for each player in the video to give you an idea of the velocities and accelerations involved for each segment. Keep in mind that I rounded to the nearest .01 and some of the discrepancies are due to the frame counting at each interval which may vary in some instances. You will not that it appears that velocity is constant or close to it from the 4045 yard segment onward. That means that all acceleration is complete at the 40 yard mark. Most football players are also at peak velocity (acceleration = 0) by the 40 yard mark as well.
Also here are some actual split times of real football players for you to play with. Note that they are all examples from the previous posts.
1.40
2.41
4.24
1.50
2.63
4.43
1.58
2.61
4.43
2.19
3.53
6.17
1.47
2.34
4.16
2.11
3.59
6.33
Thanks again and let me know if you come up with anything useful to help me be as realistic as possible!
DB
Last edited by dcebb2001; Apr 23rd 2014 at 04:01 PM.

 
Apr 24th 2014, 07:57 AM

#6  Physics Team
Join Date: Jun 2010 Location: Morristown, NJ USA
Posts: 2,347

I really have nothing further to add. I think the fact that those two players ended up with the same total rating and that they reached the 40 yard mark at the same time is a fluke. Either that or the weighting is not really from 199 for each phase. One thought I had is that if the first phase is actually scored from 4899 and the second from 199 that would make the math work  perhaps that's what they did here.
Perhaps a rather extreme example would make this clear. Suppose you had a player who could run at 200 MPH, but at the start he just waits for 4 seconds after the gun goes off, then runs the full 40 yards in 0.43 seconds to have an elapsed time of 4.43 (same as your players A and B previously). This guy would score the absolute worst for the first ten yards, so by your technique gets a score of 1 in the ACC phase, and has the absolute best time for the last 30 yards and so gets SPD =99. Even though he ties players A and B over the full 40 yards he gets a lower total. Player A now has ACC=95 and SPD=34, and player B has ACC=92 and SPD=36.

 
Apr 24th 2014, 09:06 AM

#7  Junior Member
Join Date: Dec 2012
Posts: 24

Thanks again Chip. That makes sense. I would like to leave the topic open so that anyone with any ideas on a better way I can combat the real overall issue in Post #5 could throw them out there.
The big thing I am trying to do: rate players as they are rated in Madden given real data that measures a players acceleration and speed.

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